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Iterative Data Detection and Channel Estimation for

Single-Parity Check-Product Coded MIMO Wireless

Communications System

Muladi*, N. Fisal, S. K. Yusof

Department of Telematics and Optic Communications

Fac. of Electrical Eng., Universiti Teknologi Malaysia,

Johor Bahru, Malaysia

email: *muladi@gmail.com

Abstract: In iterative data-detection and channel-estimation

algorithms, the channel estimator and the data detector

recursively exchange the information order to improve the

system performance. In this paper, maximum a posteriori based

iterative data detection and pilot symbol assisted channel

estimation of the single parity check product code for multipleantenna wireless communication is studied. Results show the

algorithm can converge at a few iteration numbers and improve

error probability performance of the system.

Index Terms:multiple antennas, single-parity check codes, product

code, iterative decoding, pilot symbol assisted channel estimation.

I.INTRODUCTION

The joint iterative data-detection and channel estimationappears to be a suitable means to achieve excellentperformance in wireless communication system, where thelength of the training sequence or pilot symbols is to be keptas small as possible to maintain the data throughput is still

high. Basically, iterative schemes (which are usually denotedthrough the adjective turbo), the channel estimator and thedata detector recursively exchange information in order toimprove the system performance.

The explosive growth of wireless communication services,such as mobile computing and wireless Internet, as well as thedeployment of wireless local area network (WLAN), hasresulted in an in bandwidth-efficient high-data-ratetransmission system. Recent results from information theoryhave shown that capacity of a multiple antennas wirelesscommunication system operating in a rich scatteringenvironment grows with a law approximately linear inminimum between the number of transmit and receive

antennas [1]. Likewise, high-performance space-time diversitysystems have been recently introduced, which permit thesystem to achieve huge performance gains with respect tosingle-antenna communication system (see. e.g [2], [3], and[4]).

Some layered space-time architectures have been proposedin order to exploit the benefit of the multiple antenna systempromise in theory and their potential gains over single antennasystems. Among these, the most popular one has been termedBell Labs Layered Space-Time Architecture (BLAST) [5].This system has attracted much attention in the last ten years

and several papers have appeared in the open literature,presenting theoretical finding and/or performance results forBLAST-like system.

In our previous work [6], the full-rate space-time diversity

scheme have been proposed. Compare to the system usingspace-time block codes [3], the proposed scheme hasadvantage that guarantee the transmission rate will alwaysequal to 1 symbol/Hz/s. This paper extends these previousresults by increasing the bit rate and develops the channelestimation. One alternative in increasing bit rate uses high-ratecode as component code of the product code but it provideslow error rate.

The work in this paper uses the single-parity-check code(SPC) as component code, parallel concatenated, of productcode (PC). SPC is simple, but weak, algebraic code. The SPC-PC codes are high-rate codes when reasonable block lengthsare used. They are simple to encode and decode and have been

shown to produce good performance. Maximum a posteriori(MAP) decoded multidimensional SPC-PC were shown tohave very good performance in [7].

By taking the output of the SPC-PC encoder into twodefined-interleavers and transmitting through multipletransmit antennas, will provide space and time diversity in thesystem. Pilot-symbol aided channel estimation is used tomeasure the channel impulse response. Following the iterationof the decoding process, the channel impulse response will beupdated by employing the extrinsic information output of thedecoder. The convergence of this channel impulse responseupdate will investigate.

II.SYSTEM MODEL

Figure 1 depicts the discrete time model of proposedsystem diagram. At the transmitter, the data stream isformatted into a block data with size ofkrxkc, where kr is thedata length in rows and kl is the data length in column. Eachrow and column of data is encoded using single parity checkcode (SPC) to be a (kr+1,kr,2) for row and (kc+1,kc,2) forcolumn. The encoding process in rows and columns are to beindependent thus the check on check parity bit is not generated[7]. Without lost of generality, in this paper it sets kr= kc = k.

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The encoded data stream output to two systematicinterleavers that defined by the row-wise reading and column-wise reading (see [6]). These data streams are then mappedinto S-symbol of constellation elements, s. The L number ofpilot symbols p are inserted at each l data information toprovide the channel response measurement. The resultedblocks have the following format:

1 1 2 1 2 1 3[ .... .... ]l l l L

l l

x p s s p s s p p+ += (1)

These blocks are transmitted through each branch oftransmit antenna. The distance Each transmit antenna uses thesame amount of energy and that totally equal to transmittedenergy of the single transmit antenna system. It is assumedthat the transmit antennas are far-separated enough thus eachpath between transmit and receive antennas are independent.

a)

b)

Figure 1 Proposed System diagram, a) transmitter, b) receiver.

The receiver employs a number of antennas that equal tothe number of transmit antennas at the transmitter. Thereceived complex base-band samples at the received antenna nare presented by:

2

,

1

n m n m n

m

r h x w=

= + (2)

where xm is transmitted block symbols from transmitantenna m. While wn is an additive noise, Gaussian distributed

with mean zero and variance No /2 when the transmitted bitenergy isEb/No.

Assuming that the transmit antennas are separated enoughas well as the receive antennas, thus the link between eachtransmit and each receive are independent. This link is in arich scattering environment that consists of multiple paths butno line of sight component and modeled as a wide-sense-stationary complex Gaussian process with zero mean, whichmakes the marginal distributions of the phase and amplitude atany given time uniform and Rayleigh, respectively, hence theterm Rayleigh fading.

III.DATA RECOVERY AND ITERATIVE DETECTION

One block of data information are interleaved by twointerleavers and transmitted from two transmit antennas. Byusing explicit notation, (2) can be rewritten as:

1 11 1 21 2 1r h x h x w= + +

2 12 1 22 2 2r h x h x w= + + (3)

The estimated received symbols are obtained bycombining the received signals,

11

* *

1 1 12 2x h r h r = +

21

* *

2 1 22 2x h r h r = +

(4)

Where (.)*

is conjugate operator. Soft-output de-mappingprocess yields the estimated bit sequences as follow:

1

1 1 ( )b M x=

12 2 ( )b M x

= (5)

Where ()-1 is demapping. Remember that the transmitted bitsequence from transmit antenna one is same with thetransmitted bit sequence from antenna two, but interleave atdifferent position prior to mapping process and transmission.

The decoding process starts by calculating the log-likelihood ratio (LLR) of two received sequences,

1

1

1

( 1 )( ) log

( 1 )e

P d bL d b

P d b

= + = =

(6)

2

2

2

( 1 )( ) log

( 1 )e

P d bL d b

P d b

= + = =

(7)

Where drepresents the transmitted data bits. The computationof (7) and (8) are same, thus it can be presented in general.Using Bayesian rule, (7) and (8) to be:

( 1) ( 1)( ) log( 1) ( 1)

( 1) ( 1)log log

( 1)( 1)

( ) ( )

e

e e

c

p b b P d L d b

p b b P d

p b d P d

P dp b d

L b L d

= + = +=

= =

= + = += + = =

= +

(8)

Assuming all bits are equal likely, the second term can beignored. This will be used as the channel LLR Lc(b). Forstatistically independent transmission of the dual diversitysystem, the LLR channel is [7]:

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1 2 ( ) ( ) ( ) ( )c c cL d L b L b L d = + + (9)

where ( )cL d is simplified version of( )L d b . The LLR

output (soft output) of the decoder is equal to [8]:

( ) ( ) ( )c eL d L d L d

= + (10)

where ( )eL d is extrinsic LLR obtained from row and

column decoding. From [6] and [8], the extrinsic LLR is:

( )( ) ( )( )

( )( ) ( )( )1, 1,

1, 1,

( )

( 1) exp ( ) 1 ( 1) exp ( ) 1

log ( 1) exp ( ) 1 ( 1) exp ( ) 1

e j

k k

i i

i i j i i j

k k

i i

i i j i i j

L d

p L d p L d

p L d p L d

= =

= =

=

+ + +

+ +

(11)

where:

( )( 1) exp ( ) 1cp L p + = +

( )( 1) exp ( ) 1cp L p = (12)

where Lc(p) is LLR channel of the parity bit at the samerow or column of the dj. At the last iteration, the soft output ofthe decoder is:

( ) ( ) ( ) ( )j c j er j ec jL d L d L d L d = + + (13)

The hard decision value of this data bit can be obtained byapplying the sign function to (13).

IV.CHANNEL ESTIMATION AND PILOT SEQUENCE DESIGN

Equation (3) can be represented in matrix form as follows:

= +R HX W (14)

This relation gives one possible way to measure thechannel response. The transmitted blockXconsists of a fewnumber of pilot symbol, xp, that the receiver knows the pilotvalue and its position within the block. A particular pilotsymbol will help to measure the channel impulse response and

the pilot positions will define the time of those responseoccurrences. The channel impulse responses will be obtainedby inverting the pilot symbol matrix, Xp, that the elements ofthis matrix are pilot symbol from antenna one and two.

11 12

2

21 22

p

p p

p p

=

X (15)

This matrix define the minimum number of pilot symbolthat should be inserted into the transmitted data block, thus thechannel impulse response measurement can be performed.

In order simplified computation of the channel impulseresponse, this paper sets the pilot symbol are orthogonal intime. In particular, the simplest pilot symbol matrixes are:

11 12

22 21

0 0

0 0

p por

p p

Once the channel impulse responses at particular time areachieved, the channel response at a whole transmitted blockcan be obtained by using interpolation, where the number ofpilot symbol define the accuracy of the generated samples.

( )1= H X R W (16)

As described at previous section, the iterative datadetection results the extrinsic information of data that willupdate the received bits value (Lc + Le). Soft-mapping these bitsequence to the constellation elements of S (soft-values). Byusing equation in (5), the new channel impulse response can

be calculated. Thus the channel impulse responses are alsoupdated iteratively together with the data detection process.

V.SIMULATION RESULT AND DISCUSSION

Simulation results are now presented for the proposedscheme. It begins with comparing the error rate performanceof the proposed system with single antenna system. The PCuses SPC with k = 8 and the flat Rayleigh fading channel isassumed. Figure 2 is shown that the proposed system has 2 dBpower advantageous than single antenna system to achieve biterror rate (BER) at 2x10-5. Each decoding iteration gaveimprovement on BER. A significant improvement is obtainedafter iteration number 3. After that the iteration is not improvethe BER performance significantly or no improvement. It has

been investigated in [9], that the effective iteration number of2 dimensional SPC-PC is 3. The curve shows at E b/N0 6 dB,the second iteration gives 1x10

-5error probability less than the

second iteration, while less error probability for 6x10-5 givesby the third iteration, and 1x10

-5gives by the forth iteration.

Figure 3 shows error rate performance of the proposedsystem with different number of pilot symbol within one blockof data information. The channel is a Rayleigh distributed flatfading and is assumed to be invariant during one symbolperiod. Number of pilot symbols compare to the number ofsymbols in one data block is presented in percentage. For thenumber of pilot symbols equal to data symbols (50%), givesBER 2x10-5 at Eb/N0 6 dB, while perfect channel estimationgives 3x10-6. This error performance decreases significantly asdecreasing the number of pilot symbols. If half number ofpilot symbols is employed, the error performance willdecrease to 2x10-5. Furthermore, the error rate performancewill decrease to 1x10-4 if the number of pilot symbol is aquarter of transmitted block length, and 2x10-3 if the numberof pilot is one over eight of the length of transmitted block.These results confirmed that higher number of pilot symbolswill give higher error rate performance in the cost of lowerrate transmission.

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Figure 2 Performance of the proposed system when the channel is perfectlyestimated.

Figure 3 Performance of the proposed system employing a numbers of pilotsymbols compared to the length of the transmitted block.

Figure 4 Convergence curve of the iterative data decoding and channelestimation; dash-line: without channel response update, full-line: with channel

response update.

Next, we investigate convergence of the iterative decodingalgorithm when the channel is not perfectly measured. As it isshown in Figure 4, the decoding algorithm will converge after5 iterations. Since more accurate information of channel isavailable, then the algorithm will converge faster. In this case,the information of the channel is brought by pilot symbol inthe transmitted block. The algorithm will converge at seconditeration when the number of pilot is equal to the number ofdata symbol in the transmitted block (50% of block length).Higher number of iteration is required to reach the algorithmconvergence when employ a few number of pilot symbols.Three and five iterations are required when the number ofpilot symbols is a quarter and one over eight of the transmittedblock, respectively. Although these number iteration are stillreasonable but the error rate performances are low.

VI.CONCLUDING REMARKS

The application of SPC-PC in multiple antennas systemwas introduced. Using two different interleavers based onrow-reading and column reading [6], the proposed systemprovided the space-time diversity. Decoding of the receiveddata can be performed iteratively based on MAP criterion.Performance of the proposed system is much better than singleantenna system.

Pilot symbol can employed to estimate the channelimpulse responses that required at signal processing and datadecoding in the proposed system. Accurate channel estimationwill be given by employing large number of pilot symbol inthe transmitted block. By using extrinsic information ofiterative data decoding to update the channel impulseresponse, the decoding process will converge faster.

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